The above configuration corresponds to a black string with a compactified direction of length \(L\) , which is fixed in terms of the
couplings of the theory. It is interesting to note that the black hole in the transverse section of the string is, in fact, analogous
to the charged Bañados-Teitelboim-Zanelli black hole
[1]}, where the NLSM coupling constant \(K\)  plays the role of the electric charge, and modulated for a factor that depends on the
flavor number, namely \(a_N\) . We can see that, as more flavors are considered in the theory, the compactification radius of the string becomes larger.
Note also that, in order to have a well defined compactification radius, the cosmological constant \(\Lambda \) 
must be negative.
